July 28, 2015

Tangle diagram crossings and quantum entanglement

In low dimensional topology we speak of tangles, while quantum physics speaks of entanglement. Similar words, but is there a deeper connection? Kauffman conjectured that the answer is yes (and I think he’s right, although maybe for other reasons). Glancing through arXiv this morning, I came across the following recent preprint:

Their result is what it says in the title. Namely, we comb the knot into a braid, and assign R-matrices to crossings. An R-matrix underlies a linear operator . The authors prove that if the this operator maps product states to product states, then it gives rise (via a certain “taking the normalized trace of the operator the braid gives” procedure) to a trivial quantum knot invariant. Thus, entanglement is an essential part of being a nontrivial quantum invariant. Very cool!

There’s a suggestive picture in my head. Entanglement is all about nonlocality, where two non-interacting objects (an overstrand and an understrand?) cannot be described as separate systems (crossing?), but are inseparably intertwined in that they share some sort of coordination. It’s the entanglement which allows the overstrand to “communicate” to the understrand that it is there, making it possible to construct a nontrivial quantum invariant.

I suspect there’s a lot more to this story. Well done Alagic, Jarret, and Jordan!